Question: Find one value of $x$ that is a solution to the equation: $(x^2-6)^2=-3x^2+18$ $x=$
We could solve for $x$ by expanding $(x^2-6)^2$, combining terms that are alike, and using the quadratic formula or factoring to solve for $x$. However there is a shorter and more elegant way to approach this problem. Let's use structural features to rewrite the equation in a simpler form. Note that $-3x^2+18=-3({x^2-6})$. This means that we can rewrite the equation as: $({x^2-6})^2=-3({x^2-6})$ If we let ${p}={x^2-6}$, we can see that this equation is in the form: ${p}^2=-3{p}$ Let's solve this equation in terms of ${p}$ : $\begin{aligned}{p}^2&=-3{p}\\\\ {p}^2+3{p}&=0\\\\ {p}({p}+3)&=0\\\\ {p}=0\ &\text{or} \ \ {p}=-3 \end{aligned}$ Since ${p}={x^2-6}$, let's substitute this value back into our two solutions in order to solve for $x$ : ${x^2-6}=0\ \ \ \text{or} \ \ \ {x^2-6}=-3$ When we solve ${x^2-6}=0$, we find that $x=\pm\sqrt{6}$. When we solve ${x^2-6}=-3$, we find that $x=\pm\sqrt{3}$. In conclusion, the four solutions of the equation $(x^2-6)^2=-3x^2+18$ are: $x=\sqrt{6}$ $x=-\sqrt{6}$ $x=\sqrt{3}$ $x=-\sqrt{3}$ [Is there another way to solve for x?]